To better understand the mechanics of the proposed layered architecture, notice that each layer \(k\)  as described in (REF )-() is characterized by five vectors \({\mathbf {a}}^{(k)}, {\mathbf {b}}^{(k)}, {\mathbf {u}}^{(k)}, {\mathbf {w}}^{(k)}, {\mathbf {v}}^{(k)} \in {\mathbb {R}}^M\) .
If we set \({\mathbf {a}}^{(k)} = \mathbf {1}\)  and \({\mathbf {b}}^{(k)} = \mathbf {0}\) , then expressions ()-() correspond to the closed-form solutions of a block-coordinate algorithm designed to solve (REF ) [1]}. Putting it differently, for these values of \({\mathbf {a}}^{(k)}\)  and \({\mathbf {b}}^{(k)}\) , our proposed architecture boils down to the classical WMMSE truncated to \(K\)  iterations.
In this setting, \({\mathbf {u}}^{(k)}\)  and \({\mathbf {v}}^{(k)}\)  represent receiver and transmitter side variables, respectively, in the sense that \(u_i^{(k)}\)  depends exclusively on the channel states into receiver \(r(i)\)  whereas \(v_i^{(k)}\)  depends exclusively on the channel states out of transmitter \(i\) .
Moreover, upon convergence, these represent measures of the strength of signals at the receiver and transmitter for our single-input single-output (SISO) system.
Their equivalent variables in the MIMO case would correspond to receiver and transmitter-side beamformers; see [1]} for additional details.
Notice also that the update of \({\mathbf {w}}^{(k)}\)  in () is local in the sense that \(w_i^{(k)}\)  only depends on the \(i\) th entries of the other vectors of interest and the channel state \(h_{ii}\)  between transmitter \(i\)  and its intended receiver \(r(i)\) .
